Search

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and which commutes with a quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a “hyperbolic” composition operator.

We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1, Comment: Change of title. New abstract and new introduction

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

We establish interpolation formulae for operator spaces that are components of a given quasi-normed operator ideal. Sometimes we assume that one of the couples involved is quasi-linearizable, some other times we assume injectivity or surjectivity in the ideal. We also show the necessity of these suppositions.

Let $${\mathscr {H}}$$ be a Hilbert space. Suppose that $$A\in {\mathbb {B}}({\mathscr {H}})$$ and the operators $$I+mA$$ are invertible for all integers $$m \ge 1$$ . We characterize the resolvent algebra $$\begin{aligned} R_A:= \left\{ T \in {\mathbb {B}}({\mathscr {H}}) : \sup _{m \ge 1}\Vert (I+mA)T(I+mA)^{-1}\Vert < \infty \right\} , \end{aligned}$$ when A is a finite rank operator with $$\mathrm{cov}(A)\ne 0$$ . Moreover, we determine the elements of $$\{A\}'$$ and $$R_A^{c_0}$$ and prove that $$R_A = R_A^c = \{A\}'\oplus R_A^{c_0}$$ , where $$\{A\}'$$ is the commutant A and both $$R_A^c$$ and $$R_A^{c_0}$$ are subclasses of $$R_A$$ defined by $$\begin{aligned} R_A^c = \left\{ T \in R_A : \lim _{m \rightarrow \infty } \Vert (I+mA)T(I+mA)^{-1}\Vert ~ \mathrm {exists} \right\} \end{aligned}$$ and $$\begin{aligned} R_A^{c_0} = \left\{ T \in R_A^c : \lim _{m \rightarrow \infty } \Vert (I+mA)T(I+mA)^{-1}\Vert =0 \right\} . \end{aligned}$$ We provide a counterexample showing that $$R_A^c= \{A\}'\oplus R_A^{c_0}$$ is not true for some compact operators.

1. Preliminaries: Set Theory and General Topology 2. Banach Spaces 3. Hilbert Spaces 4. Banach Algebras 5. Spectral Representation of Operators on Hilbert Spaces 6. The Numerical Range 7. Nonnormal Classes of Operators 8. Conditions Implying Normality 9. Symmetrizable Operators: Generalizations and Applications 10. Invariant Subspaces and Some Structure Theorems 11. The Weyl Spectrum of an Operator 12. Analytic and Quasi-Analytic Vectors 13. Schwarz Norms 14. Maximum Theorems for Operator-Valued Holomorphic 15. Uniform Ergodic Theorems for Some Classes of Operators

We extend and generalize some results in local spectral theory for upper triangular operator matrices to upper triangular operator matrices with unbounded entries. Furthermore, we investigate the boundedness of the local resolvent function for operator matrices.

Let u be a nonconstant inner function. In this paper, we study the dual truncated Toeplitz operators on the orthogonal complement of the model space K u 2 . This is a new class of Toeplitz operator. We show the product of two dual truncated Toeplitz operators D f D g to be zero if and only if either f or g is zero. We give a necessary and sufficient condition for the product of two dual truncated Toeplitz operators to be a finite rank operator. Furthermore, a necessary and sufficient condition is found for the product of two dual truncated Toeplitz operators to be a dual truncated Toeplitz operator. The last two results are different from the classical Toeplitz operator theory.

We study the Bishop-Phelps-Bollob\'as property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a~uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop-Phelps-Bollob\'as theorem holds for operators from $\ell_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute., Comment: 26 pages

The purpose of this present effort is to define a new fractional differential operator Tzβ,τ,γ, involving Srivastava–Owa fractional derivative operator. Further, we investigate some geometric properties such as univalency, starlikeness, convexity for their normalization, we also study boundedness and compactness of analytic and univalent functions on weighted μ-Bloch space for this operator. The method in this study is based on the generalized hypergeometric function. Keywords: Analytic functions, Univalent functions, Srivastava–Owa fractional derivative operator, Generalized differential operator, Weighted μ-Bloch space, Convolution (or Hadamard product)

We study the Bhatia–Semrl property which is related to Birkhoff–James orthogonality on operator spaces. Recently, many authors obtained some conditions for operators to have Bhatia–Semrl property. Using them, we explore the denseness of operators with Bhatia–Semrl property or without Bhatia–Semrl property on some classes of Banach spaces.

This paper gives necessary and sufficient conditions for an essentially normal operator T to have its C ⁎ -algebra C ⁎ ( T ) generated by a complex symmetric operator. Also it is completely determined when C ⁎ ( T ) is ⁎-isomorphic to a C ⁎ -algebra singly generated by complex symmetric operators. These both depend only on the singular part of T.

In this work we introduce the disjointly improjective operators between Banach lattices. We investigate this class of operators. Also, we extend the Flores–Hernandez’s theorem on the domination problem by disjoint strictly singular operator.

The aim of this study is to determine the systems that preserve the minimum phase property in signal processing. The minimum phase signals are closely related to outer functions. A basic mathematical question that arises in geophysical imaging is to characterize the linear operators preserving the set of outer functions in Hardy spaces. It is shown that a bounded linear operator on the Hardy space H p , 1 p ∞ , preserving the set of outer functions is necessarily a weighted composition operator. Moreover, an operator preserving the set of shifted outer functions is necessarily a weighted composition operator as well. These results complement work by Gibson and Lamoureux.

The aim of this paper is to present a comprehensive study of operator m-convex functions. Let m ∈ [ 0 , 1 ] {m\in[0,1]} , and J = [ 0 , b ] {J=[0,b]} for some b ∈ ℝ {b\in\mathbb{R}} or J = [ 0 , ∞ ) {J=[0,\infty)} . A continuous function φ : J → ℝ {\varphi\colon J\to\mathbb{R}} is called operator m-convex if for any t ∈ [ 0 , 1 ] {t\in[0,1]} and any self-adjoint operators A , B ∈ 𝔹 ⁢ ( ℋ ) {A,B\in\mathbb{B}({\mathscr{H}})} , whose spectra are contained in J, we have φ ⁢ ( t ⁢ A + m ⁢ ( 1 - t ) ⁢ B ) ≤ t ⁢ φ ⁢ ( A ) + m ⁢ ( 1 - t ) ⁢ φ ⁢ ( B ) {\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)} . We first generalize the celebrated Jensen inequality for continuous m-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operator m-convexity, we extend the Choi–Davis–Jensen inequality for operator m-convex functions. We also present an operator version of the Jensen–Mercer inequality for m-convex functions and generalize this inequality for operator m-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operator m-convex functions, we construct the m-Jensen operator functional and obtain an upper bound for it.

We present necessary and sufficient conditions that a family A = { a i j : i , j ∈ I } of real numbers may be considered as a bounded linear operator on Banach spaces l 1 ( I ) and l ∞ ( I ) , where I is an arbitrary non-empty set. Moreover, we get that these conditions are sufficient for a family to be a bounded linear operator on l p ( I ) , for each p ∈ [ 1 , ∞ ] . Within this class of operators, the notion of doubly superstochastic operator is introduced as an extension of the doubly superstochastic matrix, and some of its essentially properties are presented. In the second part, we extend the notion of weak supermajorization relation on the Banach space l p ( I ) using doubly superstochastic operators, and present close relationship between this relation and superstochastic operators as generalisation well-known results in the theory of majorization. Among others, for two functions f , g ∈ l 1 ( I ) + we show that relations f ≺ w s g and g ≺ w s f hold if and only if there exist a permutation P such that g = P f .

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is “the generalized Neumann lemma” which is quite different from the method in [12] where “the generalized Banach lemma” was used. By the method of the perturbation analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

We establish the conditions of bifurcation of the solutions of weakly perturbed operator equations in Banach spaces from the point 𝜀 = 0 and propose a convergent iterative procedure for finding the solutions in the form of segments of powers series in 𝜀 with pole at the point 𝜀 = 0.

Let C 1 , C 2 , … , C k be positive matrices in M n and f be a continuous real-valued function on [ 0 , ∞ ) . In addition, consider Φ as a positive linear functional on M n and define ϕ ( t 1 , t 2 , t 3 , … , t k ) = Φ ( f ( t 1 C 1 + t 2 C 2 + t 3 C 3 + … + t k C k ) ) , as a k variables continuous function on [ 0 , ∞ ) × … × [ 0 , ∞ ) . In this paper, we show that if f is an operator convex function of order mn, then ϕ is a k variables operator convex function of order ( n 1 , … , n k ) such that m = n 1 n 2 … n k . Also, if f is an operator monotone function of order n k + 1 , then ϕ is a k variables operator monotone function of order n. In particular, if f is a non-negative operator decreasing function on [ 0 , ∞ ) , then the function t → Φ ( f ( A + t B ) ) is an operator decreasing and can be written as a Laplace transform of a positive measure.

Several new pre-orders are introduced and characterized on the set of all wg -Drazin invertible elements of a Banach algebra. We generalize recent results for rectangular matrices and bounded linear operators between Banach spaces, omitting some assumptions and adding new characterizations.

In this paper, we extend the Roper-Suffridge extension operator in complex Banach space, and prove that the extended Roper-Suffridge operators preserve the properties of the subclasses of spirallike mappings on the unit ball in complex Banach spaces. Thereby, we promote the conclusions to the cases in complex Hilbert spaces. The conclusions provide new approaches to construct these subclasses of spirallike mappings in several complex variables.

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.

In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class $$C_{0}.$$ With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction.

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators. Suppose $F(\cdot )$ is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the $C$-numerical range, or the $C$-numerical radius for some finite rank operator $C$. The structure is determined for surjective maps $\Phi :{\mathcal A}\rightarrow \mathcal B$ satisfying $F(A^*B)=F(\Phi (A)^*\Phi (B))$ for all $A, B \in {\mathcal A}$. To establish the proofs, some general results are obtained for functions $F:{\mathcal F}_1(H) \cup \{0\} \rightarrow [0, +\infty)$, where ${\mathcal F}_1(H)$ is the set of rank one operators in ${\mathcal B}(H)$, satisfying (a) $F(\mu UAU^*)=F(A)$ for a complex unit $\mu$, $A\in {\mathcal F}_1(H)$ and unitary $U \in {\mathcal B}(H)$ (b) for any rank one operator $X\in {\mathcal F}_1(H)$ the map $t\mapsto F(tX)$ on $[0, \infty)$ is strictly increasing, and (c) the set $\{F(X): X \in {\mathcal F}_1(H) \hbox{ and } \|X\| = 1\}$ attains its maximum and minimum., Comment: 14 pages

This paper is concerned with the self-adjoint perturbations of the spectra for the upper triangular partial operator matrix with given diagonal entries. A necessary and sufficient condition is given under which such operator matrix admits a Weyl (Fredholm) operator completion by choosing some bounded self-adjoint operator. It is shown that the self-adjoint perturbation of the Weyl (essential) spectrum can be the proper set of the general perturbation. Combining the spectral properties, we further characterize the perturbation of the Weyl (essential) spectrum for Hamiltonian operators.

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution ⋄ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if F j , G j ( 1 ≤ j ≤ N ) are polynomials of z and z ¯ then ∑ j = 1 N T F j T G j − T H is a finite rank operator for some L 1 -function H if and only if ∑ j = 1 N F j ⋄ G j belongs to L 1 and H = ∑ j = 1 N F j ⋄ G j . In the case F j 's are holomorphic and G j 's are conjugate holomorphic, it is shown that H is a solution to a system of first order partial differential equations with a constraint.

We prove the existence and uniqueness of a solution to the one-point initial problem for the linear differential–difference equation u′(z) = Au(z + h), h ϵ ℂ, in some classes of exponential type entire functions. We obtain a representation of a unique solution to this problem by using the operator Bruwier series.

In this paper we study the essential spectrum of the operator LAu(x)=−(A(x)u′(x))′+u(x)where A(x) is a positive absolutely continuous function on (0, 1) that resembles x2I± for some I±â‰¥1. We prove that the essential spectrum of LA coincides with the essential spectrum of the operator LI±u(x):=−(x2I±u′(x))′+u(x).

In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.

In this work, we introduce implicit and explicit iterations for solving the variational inclusion problem for the sum of two operators and the fixed point problem of nonexpansive mappings. We then prove its strong convergence theorems in the framework of Banach spaces. We finally provide some applications of the main results.

In this paper, we study the bounded approximation property for the weighted space$\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subsetUof a Banach spaceEand its predual$\mathcal{GV}$(U), where$\mathcal{V}$is a countable family of weights. After obtaining an$\mathcal{S}$-absolute decomposition for the space$\mathcal{GV}$(U), we show thatEhas the bounded approximation property if and only if$\mathcal{GV}$(U) has. In case$\mathcal{V}$consists of a single weightv, an analogous characterization for the metric approximation property for a Banach spaceEhas been obtained in terms of the metric approximation property for the space$\mathcal{G}_v$(U).

Let α ∈ ( 0 , n - 1 2 ) and Tα be the Bochner-Riesz operator of order α. In this paper, for n = 2 and n ≥ 3, the compactness on Lebesgue spaces and Morrey spaces are considered for the commutator of Bochner-Riesz operator generated by CMO(ℝn) function and Tα.

We develop a general theory of order isomorphisms of operator intervals. In this way we unify and extend several known results, among others the famous Ludwig’s description of ortho-order automorphisms of effect algebras and Molnar’s characterization of bijective order preserving maps on bounded observables. Besides proving several new results, one of the main contributions of the paper is to provide self-contained proofs of several known theorems whose original proofs depend on various deep results from functional analysis, operator algebras, and geometry. At the end we will show the optimality of the obtained theorems using Lowner’s theory of operator monotone functions.

The spectra of the generalized difference operator B(r, s) in various sequence spaces have been investigated by many authors. Nevertheless, to the best of our knowledge, no contribution has appeared so far to study the problem in the common sequence spaces and and . In this paper, we fill this gap by analysing the spectrum of the operator B(r, s) on and h. Also, we explore some ideas of how to study the problem for a general form of the operator, namely, the operator .

Let be a bounded domain of class . In , we study a self-adjoint strongly elliptic operator of order 2p given by the expression , , with Neumann boundary conditions. Here, is a bounded and positive definite matrix-valued function in , periodic with respect to some lattice; is a differential operator of order p. The symbol is subject to some condition ensuring strong ellipticity of the operator . We find approximations for the resolvent in different operator norms with error estimates depending on and .

Let X be a quasi-Banach space, Y be a γ-Banach space (0

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide. Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK); Research Foundation of Namik Kemal UniversityNamik Kemal University [NKUBAP.01.GA.17.108] Author E. Bayram was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) in the context of 2219-Post Doctoral Fellowship Program and by the Research Foundation of Namik Kemal University (Project No. NKUBAP.01.GA.17.108).

Let A and B be two positive operators with for positive real numbers be an operator mean and be the adjoint mean of If and is a positive unital linear map, thenwhereand is the Kantorovich constant. In addition, for

We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and whose weights are multiplication operators in L 2 -spaces, under the assumption of existence of a family of probability measures whose Radon–Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension, which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators.

A well-known result of J. Lindenstrauss and A. Pelczynski (1968) gives the existence of a universal non-weakly compact operator between Banach spaces. We show the existence of universal non-Rosenthal, non-limited, and non-Grothendieck operators. We also prove that there does not exist a universal non-Dunford–Pettis operator, but there is a universal class of non-Dunford–Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non-weakly compact polynomials, there does not exist a universal polynomial.

We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). Defining the gauge of an operator $T \in B(H)$ to be $\|Re(T)_+\|$, we demonstrate an abstract characterization of operator spaces up to complete gauge-isometry. Both of these characterizations preserve the structure of the self-adjoint, positive, and accretive operators, as well as the operator norm. We show that an operator space with a given matrix ordering of positive or accretive cones can be represented completely isometrically and completely order isomorphically if and only if each positive cone is normal, in the sense that $x \leq y \leq z$ implies that $\|y\| \leq \max(\|x\|,\|z\|)$ at each matrix level. This is achieved by showing that normal matrix ordered operator spaces are induced by gauges. We show that inducing gauges are not unique in general. Finally, we show that completely positive completely contractive linear maps on non-unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge., Comment: 18 pages. Exposition significantly updated thanks to helpful comments from the editors and referees. To appear in the Journal of Mathematical Analysis and Applications

In this article, an equivalent condition for the existence of a bounded quasi-linear (BQL) generalized inverse of a closed linear operator with respect to projector between two Banach spaces is giv...